“Helmholtz Differential Equation--Elliptic Cylindrical Coordinates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Helmholtz Differential Equation--Elliptic Cylindrical Coordinates

释义

Helmholtz Differential Equation--Elliptic Cylindrical Coordinates

In Elliptic Cylindrical Coordinates, the Scale Factors are , , and the separation functions are , giving aStäckel Determinant of . The Helmholtz differentialequation is

(1)

Attempt Separation of Variables by writing

(2)

then the Helmholtz Differential Equation becomes

(3)

Now divide by

to give

(4)

Separating the

part,

(5)

(6)

which has the solution

(7)

Rewriting (5) gives

(8)

which can be separated into

			(9)
			(10)

(11)

(12)

Now use

(13)

(14)

to obtain

(15)

(16)

Regrouping gives

(17)

(18)

Let

and

, then these become

(19)

(20)

Here, (20) is the Mathieu Differential Equation and (19) is the modified Mathieu DifferentialEquation. These solutions are known as Mathieu Functions.

See also Elliptic Cylindrical Coordinates, Mathieu Differential Equation, Mathieu Function

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 657, 1953.

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